I want to define a function knowing only it's limits

In summary, defining a function using limits helps us understand its behavior and important characteristics. To determine the limit of a function, we can use methods such as direct substitution, algebraic manipulation, or graphical analysis. A function can have different limits at different points, and we can use limits to define a function by creating a piecewise function based on these limits. However, a function cannot be defined solely using its limits as other characteristics are also needed.
  • #1
maxhersch
21
0
This is a random problem I am trying to figure out. The context doesn't matter.

I wish to define a function z(x, y) based on the following limits:

1. lim z (x→∞) = 0
2. lim z (x→0) = y
3. lim z (y→∞) = ∞
4. lim z (y→0) = 0
 
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  • #2
##z(x,y)=ye^{-x}##? Can you think of another one?
 
  • #3
##z=\frac{y}{1+x}##.
 
  • #4
mathman said:
##z=\frac{y}{1+x}##.
For ##x<-1## this doesn't have the right behavior for ##y \to \infty##. If we require x,y to be non-negative it works.
 
  • #5
##z=\frac{y}{1+x^2}##
 
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1. What is the purpose of defining a function using limits?

Defining a function using limits allows us to understand the behavior of the function as it approaches a certain value, and can help us determine important characteristics such as continuity and differentiability.

2. How do I determine the limit of a function?

To determine the limit of a function, we can use various methods such as direct substitution, algebraic manipulation, or graphical analysis. It is important to consider the behavior of the function as it approaches the given value from both the left and right sides.

3. Can a function have different limits at different points?

Yes, a function can have different limits at different points. This can occur if the function has a discontinuity or if the behavior of the function changes at a specific point.

4. How do I use limits to define a function?

To use limits to define a function, we must first determine the limit of the function as it approaches a specific value. We can then use this information to create a piecewise function that describes the behavior of the original function at different intervals.

5. Can a function be defined solely using its limits?

No, a function cannot be defined solely using its limits. While limits can provide important information about the behavior of a function, they do not fully describe the function itself. Other characteristics such as the domain, range, and specific function values are also needed to fully define a function.

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